3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

3.012 Fund of Mat Sci: Structure – Lecture 20

SYMMETRIES AND TENSORS

Einstein explaining the Einstein convention

Photograph removed for copyright reasons.

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Homework for Mon Nov 28

• Study: 3.3 Allen-Thomas (Symmetry constraints)

• Read all of Chapter 1 Allen-Thomas

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Last time:

1. Atoms as spherical scatterers2. Huygens construction → Laue condition3. Ewald construction4. Debye-Scherrer experiments

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Scalars, vectors, tensors

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Scalars, vectors, tensors

1 11 1 12 2 13 3

2 21 1 22 2 23 3

3 31 1 32 2 33 3

j E E Ej E E Ej E E E

σ σ σσ σ σσ σ σ

= + += + += + +

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Einstein’s convention

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Transformation of a vector

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Transformation of a vector

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Orthogonal Matrices

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Transformation of a tensor

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Transformation law for products of coordinates

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Neumann’s principle

• the symmetry elements of any physical property of a crystal must include all the symmetry elements of the point group of the crystal

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Symmetry constraints

• Determine the crystallographic point group• Choose a generator group (set of symmetry

operation which fully generates the complete point group symmetry)

• Transform all components of a tensor by each of the symmetry elements

• Impose Neumann’s principle that a tensor component and its transformed remain identical for a symmetry operation

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Symmetry constraints

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Symmetry constraints

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Scalar, vector, tensor properties

• Mass (0), polarization (1), strain (2)

Figure by MIT OCW.

Y

Und

efor

med

X

DeformedZ

X

P P'u

R

PP'

R'

u = R' - R

o

Z

Y

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Physical properties and their relation to symmetry

• Density (mass, from a certain volume)• Pyroelectricity (polarization from temperature)• Conductivity (current, from electric field)• Piezoelectricity (polarization, from stress)• Stiffness (strain, from stress)

3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)

Curie’s Principle

• a crystal under an external influence will exhibit only those symmetry elements that are common to both the crystal and the perturbin influence